Carbene Routes to Cyclopropatetrahedrane

The formation of cyclopropatetrahedrane (tetracyclo[2.1.0.01,3.02,4]pentane) via four different carbene reactions is computed using the (U)CCSD(T)(full)/cc-pVTZ//(U)ωB97X-D/cc-pVTZ + 1.3686(EZPVE) theoretical model. Intrinsic reaction coordinate plots confirm that each carbene is directly linked to cyclopropatetrahedrane via a unique cyclopropanation step. Each elementary step is assessed according to the structure and energy of its transition state.


PLATONIC SOLIDS
Tetrahedrane (2), hexahedrane (3; cubane), and dodecahedrane (4) are fused polycyclic hydrocarbons (Table S1). S1 Their cage-like C-atom frameworks correspond to a regular (1) tetrahedron, (2) hexahedron (i.e., cube), and (3) dodecahedron, which are three of the five regular polyhedra that have been considered sacred since the time of the ancient Greeks. The syntheses of 3 and 4 are a remarkable achievement despite the structures' high symmetries and total strain energies (E s )s. S2 However, 2 has never been prepared although polysubstituted derivatives are known. S3 The six curved C-C "banana" bonds of 2 have high p orbital character and bow outside the C-C internuclear axes by 21 deg to relieve strain. The E s per C-C bond value for 2 exceeds that of 3 and far exceeds that of 4. energy, due to translations (3(½)RT) and rotations (3(½)RT), for each nonlinear molecule was then added. Finally, RT (i.e., "pV work" needed to expand one mole of ideal gas to V = 24.465 L at T = 298.15 K and p = 1 atm) was added to obtain H T (eq S1). The experimental singlet-triplet energy gap (E S -T ) of CH 2 (eq S2) S8 was used to compute the corrected E S -T of carbene 8 (eq S3).

MOLECULAR ENERGIES AND 3-D CARTESIAN COORDINATES
All ORTEP structures are shown as 50% ellipsoids.
Methylene ( 1 CH 2 ); CCSD(T)(full)/cc-pVTZ//B97X-D/cc-pVTZ +  Figure S1. Molecular orbitals (MO)s from select molecules are displayed above. (a, b) Cyclopropane has a high-lying C-C "banana" bond. The outwardly curved / bond is a hybrid between a C-C  bond and a C-C  bond. (c, d) Tetrahedrane (2) also has high-lying C-C "banana" bonds. (e, f) In (tetrahedryl)carbene (8), the vacant p orbital of the :CH-group attracts electron density from the vicinal C-C "banana" bond. This elongates and weakens the C1'-C4' bond of the tetrahedryl group. (g, h) The :CH-group's p orbital is not vacant in triplet (tetrahedryl)carbene ( 3 8). Its :CH-group does not bend toward the C1'-C4' bond because a nodal plane exists between the tetrahedryl group and the singly occupied p orbital. All structures and MOs were computed using the (U)B97X-D/cc-pVTZ //(U)B97X-D/cc-pVTZ theoretical model. All computer-generated MOs are shown with an isosurface value of 0.110 to emphasize the cores of the electron-clouds.

GEOMETRIC ANALYSIS OF INVERTED CARBON ATOMS
Below is a straightforward procedure for determining whether the four bonds emanating from a tetracoordinate carbon atom are monohemispherical. Carbene 8 is used as an example (see the original spreadsheet file for the detailed formulas).
(2) Calculate cross product u × v (in that order) to find t, which is perpendicular to both u and v.
(3) Compute  st and  rt from the dot products s • t and r • t, respectively.
(2) Calculate cross product u × v (in that order) to find t, which is perpendicular to both u and v.
(3) Compute  st and  rt from the dot products s • t and r • t, respectively.